The noncommutative geometry of the Landau Hamiltonian: Differential aspects

Abstract: In this work we study the differential aspects of the noncommutative geometry for the magnetic $C^*$-algebra which is a 2-cocycle deformation of the group $C^*$-algebra of $\mathbb{R}^2$. This algebra is intimately related to the study of the Quantum Hall Effect in the continuous, and our results aim to provide a new geometric interpretation of the related Kubo's formula. Taking inspiration from the ideas developed by Bellissard during the 80's, we build an appropriate Fredholm module for the magnetic $C^*$-algebra based on the magnetic Dirac operator which is the square root (`a la Dirac) of the quantum harmonic oscillator. Our main result consist of establishing an important piece of Bellissard's theory, the so-called second Connes' formula. In order to do so, we establish the equality of three cyclic 2-cocycles defined on a dense subalgebra of the magnetic $C^*$-algebra. Two of these 2-cocycles are new in the literature and are defined by Connes' quantized differential calculus, with the use of the Dixmier trace and the magnetic Dirac operator.